Dunkl theory is a generalization of Fourier analysis and special function theory related to root systems and reflections groups. The Dunkl operators $T_{j}$, which were introduced by C. F. Dunkl in 1989, are deformations of directional derivatives by difference operators related to the reflection group. The goal of this talk will be to study harmonic analysis in the rational Dunkl setting. The first part will be devoted to some of results obtained in recent joint works with Jacek Dziubanski (2019, 2020).

• improved estimates of the heat kernel $h_t(\mathbf{x},\mathbf{y})$ of the Dunkl heat semigroup generated by Dunkl–Laplace operator $\Delta_k=\sum_{j=1}^{N}T_j^2$ expressed in terms of analysis on the spaces of homogeneous type;

• theorem regarding the support of Dunkl translations $\tau_{\mathbf{x}}\phi$ of $L^2$ compactly supported function $\phi$ (not necessarily radial).

The results listed above turn out to be useful tools in studying harmonic analysis in Dunkl setting. We will discuss this kind of applications in the second part of the talk. We will focus on a version of the classical Hormander’s multiplier theorem proved in joint work with Dziubanski (2019). If time permits, we will discussed how our tools can be used to for studying singular integrals of convolution type or Littlewood–Paley square functions in the Dunkl setting.